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eigenvalue (Definition)

Let $ V$ be a vector space over a field $ k$, and let $ A$ be an endomorphism of $ V$ (meaning a linear mapping of $ V$ into itself). A scalar $ \lambda\in k$ is said to be an eigenvalue of $ A$ if there is a nonzero $ x \in V$ for which

$\displaystyle Ax = \lambda x\;.$ (1)

Geometrically, one thinks of a vector whose direction is unchanged by the action of $ A$, but whose magnitude is multiplied by $ \lambda$.

If $ V$ is finite dimensional, elementary linear algebra shows that there are several equivalent definitions of an eigenvalue:

(2) The linear mapping

$\displaystyle B=\lambda I - A$
i.e. $ B:x\mapsto \lambda x-Ax$, has no inverse.

(3) $ B$ is not injective.

(4) $ B$ is not surjective.

(5) $ \det(B)=0$, i.e. $ \det(\lambda I-A)=0$.

But if $ V$ is of infinite dimension, (5) has no meaning and the conditions (2) and (4) are not equivalent to (1). A scalar $ \lambda$ satisfying (2) (called a spectral value of $ A$) need not be an eigenvalue. Consider for example the complex vector space $ V$ of all sequences $ (x_n)_{n=1}^{\infty}$ of complex numbers with the obvious operations, and the map $ A: V \to V$ given by

$\displaystyle A(x_1, x_2, x_3, \dots) = (0, x_1, x_2, x_3, \dots) \;. $
Zero is a spectral value of $ A$, but clearly not an eigenvalue.

Now suppose again that $ V$ is of finite dimension, say $ n$. The function

$\displaystyle \chi(\lambda)=\det(B)$
is a polynomial of degree $ n$ over $ k$ in the variable $ \lambda$, called the characteristic polynomial of the endomorphism $ A$. (Note that some writers define the characteristic polynomial as $ \det(A-\lambda I)$ rather than $ \det(\lambda I-A)$, but the two have the same zeros.)

If $ k$ is $ \mathbb{C}$ or any other algebraically closed field, or if $ k=\mathbb{R}$ and $ n$ is odd, then $ \chi$ has at least one zero, meaning that $ A$ has at least one eigenvalue. In no case does $ A$ have more than $ n$ eigenvalues.

Although we didn't need to do so here, one can compute the coefficients of $ \chi$ by introducing a basis of $ V$ and the corresponding matrix for $ B$. Unfortunately, computing $ n \times n$ determinants and finding roots of polynomials of degree $ n$ are computationally messy procedures for even moderately large $ n$, so for most practical purposes variations on this naive scheme are needed. See the eigenvalue problem for more information.

If $ k=\mathbb{C}$ but the coefficients of $ \chi$ are real (and in particular if $ V$ has a basis for which the matrix of $ A$ has only real entries), then the non-real eigenvalues of $ A$ appear in conjugate pairs. For example, if $ n=2$ and, for some basis, $ A$ has the matrix

$\displaystyle A = \begin{pmatrix}0 & -1 \\ 1 & 0 \end{pmatrix} $
then $ \chi(\lambda)=\lambda^2+1$, with the two zeros $ \pm i$.

Eigenvalues are of relatively little importance in connection with an infinite-dimensional vector space, unless that space is endowed with some additional structure, typically that of a Banach space or Hilbert space. But in those cases the notion is of great value in physics, engineering, and mathematics proper. Look for “spectral theory” for more on that subject.



"eigenvalue" is owned by Koro. [ full author list (4) | owner history (1) ]
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See Also: eigenvalue problem, similar matrix, eigenvector, singular value decomposition

Also defines:  eigenvalue, spectral value

Attachments:
eigenvalue (of a matrix) (Definition) by mathcam
multiplicity of eigenvalue (Definition) by matte
invariance of eigenvalues (Definition) by matte
spectrum of $A-\mu I$ (Theorem) by PrimeFan
finding eigenvalues (Example) by PrimeFan
eigenvalues of normal operators (Theorem) by asteroid
example of bounded operator with no eigenvalues (Example) by asteroid
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Cross-references: Hilbert space, Banach space, structure, infinite-dimensional, conjugate, real, eigenvalue problem, variations, even, roots, determinants, matrix, basis, coefficients, eigenvalue, odd, algebraically closed, characteristic polynomial, variable, degree, polynomial, function, finite, map, operations, obvious, complex numbers, sequences, complex, dimension, infinite, surjective, injective, inverse, definitions, equivalent, linear algebra, finite dimensional, action, vector, scalar, linear mapping, endomorphism, field, vector space
There are 49 references to this entry.

This is version 11 of eigenvalue, born on 2002-01-19, modified 2006-06-09.
Object id is 1496, canonical name is Eigenvalue.
Accessed 68659 times total.

Classification:
AMS MSC: 15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors)
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